Open problems to an infinite system of quasi-linear partial differential equations with non-local terms (Symposium on Probability Theory)

Open

problems

to

an

infinite system of

quasi-linear partial

differential

equations

with

non-local terms

Tetsuya Hattori

Faculty of Economics, Keio University

1

Main results.

We consider a generalization ofa system of first order quasilinear partial differential equations

(PDE) in 1 $+$1 space-timedimensions, such that the method _ofcharacteristic

curve

_{is effective.}

We considernon-local (integration) terms and a system of infinitely manycomponents (in fact,

equations for a

measure

valued function), and focus on solutions expressed by certain

non-Markovian point process. We first summarize in this section

our

latest results in [2], and then

post openproblems in

_\S

2..

1.1

Differential

equation

with non-local

term.

Throughout this paperwe fix$T>0$. Let $W\subset C^{1}([0,1]\cross[0, T];[0, \infty))$ _{be a set of non-negative}

valued $C^{1}$ functions

on $[0$,1$]$ $\cross[0, T]$, and aBorel probability

measure

$\lambda$

supportedon the Borel

measurable space $(W, \mathcal{B}(W))$. $\mathcal{B}(W)$ is the a-algebra generated by open sets with the topology

from the space of continuous functions $C^{0}([0,1]\cross[0, T];[0, \infty))\supset C^{1}([0, 1] \cross[0, T];[0, \infty))$ _with

the metric given by the supremumnorm

(1) $\Vert w\Vert_{T}= \sup |w(y, t$

$(y,t)\in[0,1]\cross[0,T]$

For the probability space $(W, \mathcal{B}(W), \lambda)$, the Borel sets are defined by the topology of the

space of continuous functions $w:[0$,1$]$ $\cross[0, T]arrow \mathbb{R}$ with maximum norm, and we assume

(2) $M_{W}:= \int_{W}\Vert w\Vert_{T}\lambda(dw)<\infty$, and

(3) $C_{W}:= \sup_{w\in W}\Vert\frac{\partial w}{\partial y}\Vert_{T}<\infty.$

Denote the set ofinitial $(t=0)$ _{points in the space-time} $[0, 1]\cross[0, T]$, _{the set ofupper stream}

boundary $(y=0)$ points, and their union, the set ofinitial/boundary points, respectively by

$\Gamma_{b}=\{0\}\cross[0, T]=\{(0, s)|0\leqq s\leqq T\},$

(4)

$\Gamma_{i}=[0, 1 ] \cross\{0\}=\{(z, 0)|0\leqq z\leqq 1\}, \Gamma=\Gamma_{b}\cup\Gamma_{i}$

.

For$t\in[0, T]$, denote the set ofinitial/boundary points up to time $t$ by

(5) $\Gamma_{t}=\{(z, s)\in\Gamma|t_{0}\leqq t\}=\Gamma_{i}\cup\{(0, to) \in\Gamma_{b}|0\leqq t_{0}\leqq t\},$

and theset of admissible pairs of the initial/boundary point $\gamma$ and time $t$ by

(6) $\triangle\tau:=\{(\gamma, t)\in\Gamma_{T}\cross[0, T]|\gamma\in\Gamma_{t}\}.$

Let $\mu_{0}=\mu_{0}(dw\cross dz)$ be a Borel probability measure on $(W\cross[0,1],$$\mathcal{B}(W\cross[0,1$ We

Theorem 1 ([2]) There exists

a

unique pair

_{of functions}

$yc$ and$\mu\iota(dw\cross dz)$, where $yc$ is a

function of

$(\gamma, t)\in\Delta_{T}$ taking values in $[0$,1$]$, and$\mu_{t}(dw\cross dz)$ is a

function of

$t\in[0, T]$ taking

values in the probability measures on $W\cross[O$,1$]$, such that the following hold.

$yc((y0,0), t)$ is non-decreasingin$y0,$ $yc((0, t_{0}), t)$ is non-increasingin$t_{0}$, and$yc(\gamma, t)$ is

non-decreasing in $t,$ $y_{C}(\gamma, t)$ and $\frac{\partial y_{C}}{\partial t}(\gamma, t)$ are

continuous, and

_for

each $t\in[0, T],$ $y_{C}(\cdot, t)$ : $\Gamma_{t}arrow$

$[0$,1$]$ is surjective. Furthermore, $\int_{W}h(w)\mu_{t}(dw\cross[y, 1))$ is Lipschitz continuous in $(y, t)$,

for

all

bounded measurable$h:Warrow \mathbb{R}$, with Lipschitzconstant

uniform

in$h$ satisfying _{$\sup|h(w)|\leqq 1,$}

$w\in W$ and finally, the following equation

_of

motionwith initial/boundary conditions hold.

(7) $yc((y_{0}, t_{0}), t_{0})=y0, (y_{0}, t_{0})\in\Gamma,$

(8) $\mu_{t}(dw\cross[0,1))=\lambda(dw) , t\in[0, T],$

(9) $\mu_{t}(W\cross[y, 1))=1-y, (y, t) \in[0, 1 ] \cross[0, T],$

$\mu\iota(dw\cross[yc((y0, t_{0}), t), 1))=\mu_{t_{0}}(dw\cross[y0, 1))$

(10)

$- \int_{t_{0}}^{t}\int_{z\in[yc((y0,t_{0}),s),1)}w(z, s)\mu_{s}(dw\cross dz)ds,((y_{0}, t_{0}), t)\in\triangle\tau.$

$◇$

We have stated Lipschitz continuous broad solution in Theorem 1 which is a generalization of

Lipschitz solution in [1].

In terms of equation of motion for fluids, (10) is interpreted as the equation of motion of incompressible fluid mixture of infinite components in aline segment, where each component is labelled by its space-time dependent evaporation rate function $w\in W$, and the motion of the fluid mixture is driven by the evaporation. The choice of solution (with initial condition) (9)

implies incompressibility of the fluid, so that the fluid is pushed downstream in a way to keep

the density of total fluid constant. The boundary condition (8) implies conservation of fluid

components, so that the upperstream is filled by the evaporated fluid in such a way that each fluid component is conserved.

Concerning (8), since quantity offluidcomponents (i.e. the

measure

$\lambda$

)isconserved, explicit

$t$ dependence is absent if$\lambda$

is supported on functions constant in $t$

.

We will discuss, as anopen

problem in

_\S

2.2, the existence of solution $\mu_{t}$ which is constant in $t$, in such cases.

We assumed that $\mu 0$ is absolutely continuous with respect to $\lambda\cross dz$. Denote the density

function by $\sigma$:

(11) $\mu_{0}(dw\cross dz)=\sigma(w, z)\lambda(dw)dz, (w, z)\in W\cross[O, 1 ].$

Then (8) and (9) for$t=0$ respectively implies

(12) $\int_{0}^{1}\sigma(w, z)dz=1,$ $w\in W$, and

(13) $\int_{W}\sigma(w, y)\lambda(dw)=1, y\in[0, 1 ].$

Note also that asubstitution $y=yc(\gamma, t)$ in (9) implies$yc(\gamma, t)=1-\mu_{t}(W\cross[yc(\gamma, t),$$1$ with

which (10) and (9) imply

To keep the integration in (14) finite, the assumption (2) on $(W, \lambda)$ is a natural limitation. It

turned _{out that to prove Theorem 1 under this weakest} condition, it was unavoidable to obtain

an explicit formula for the solution given in

\S

1.3 in terms ofa process introduced in

\S

1.2. In

\S 2.1

we will consider relaxing the other assumption (3), and raiseas anopen problem, its effect

on uniqueness of the solution.

1.2

Point process

with

last-arrival-time

dependent

intensity.

Let $N=N(t)$ , $t\geqq 0$,be a non-decreasing,_{right-continuous, non-negativeinteger valued}

stochas-tic process with $N(0)=0$ , and for each $k\in \mathbb{Z}+$ define its k-th arrival time$\tau_{k}$ by

(15) $\tau_{k}=\inf\{t\geqq 0|N(t)\geqq k\},$ $k=1$,2, . . ., and $\tau_{0}=0.$

The arrival times $\tau_{k}$ are non-decreasing in $k$, because $N$ is non-decreasing, and since $N$ is also

right-continuous, the arrival times are stopping times; ifwe denote the associated filtration by $\mathcal{F}_{t}=\sigma[N(s), s\leqq t]$, then $\{\tau_{k}\leqq t\}\in \mathcal{F}_{t},$ $t\geqq 0.$

Let $\omega$ be a non-negative valued bounded

continuous function of $(s, t)$ for $0\leqq s\leqq t$, and for

$k=1$,2, ,..

assume

that

(16) $P[t<\tau_{k}|\mathcal{F}_{\tau_{k-1}}]=\exp(-\Omega(\tau_{k-1}, t))$ on $t\geqq\tau_{k-1},$

where, for $t\geqq t_{0}$ put

(17) $\Omega(t_{0}, t)=\int_{t_{0}}^{t}\omega(t0, u)du.$ If$\omega$is independent of the first

variable, then (16) implies that$N$is the (inhomogeneous) Poisson

process with intensity function $\omega$. We are considering a generalization of the Poisson process

such that the intensity function depends on the latest arrival time.

A construction of thepointprocess withlast-arrival-timedependent intensity goesasfollows. Let $\omega$ : _{$[0, \infty)^{2}arrow[0, \infty)$} be a

non-negative valued bounded continuous function of $(s, t)$ _for

$0\leqq s\leqq t$_, for _which we _{aim to construct a process satisfying (16). Let} $\nu$ be a Poisson random

measure

on $[0, \infty)^{2}$, with unit constant intensity

$E[\nu([a, b]\cross[c, d])]=(b-a)(d-c) , b>a>0, d>c>0, k\in \mathbb{N}.$

Define a sequence of hitting times $\tau_{k},$ $k\in \mathbb{Z}+$, inductively by $\tau_{0}=0$, and, for _$k=1$,2,.

.

.,

(18) $\tau_{k}=\inf\{t\geqq\tau_{k-1}|\nu(\{(\xi, u)\in[0, \infty)^{2}|0\leqq\xi\leqq\omega(\tau_{k-1}, u), \tau_{k-1}<u\leqq t\})>0\}.$

$\tau_{k}$ in (16), is defined by (18), and the process $N(t)$ is defined by the reciprocal relation to

(15): $N(t)= \max\{k\in \mathbb{Z}_{+}|\tau_{k}\leqq t\},$ $t\geqq$ O. $\{\tau_{k}\leqq t\}isin\mathcal{F}_{t}:=\sigma[\nu(A);A\in \mathcal{B}([0, \infty)^{2})$, $A\subset$

$[0, \infty)\cross[0, t],$ $k\in \mathbb{N}]$, and consequently $N$ is adapted to $\{\mathcal{F}_{t}\}$. Basic formulas, for $N(t)$ to be

used in a proof of the main theorem, are in [3].

1.3

Flows and

construction

of solution. Definethe set of flows $\Theta_{T}$ on $[0$, 1$]$ $\cross[0, T]$ by

$\Theta_{T}$ $:=\{\theta$ : $\Delta_{T}arrow[0$,1$]$ $|\theta((y_{0}, t_{0}), t_{0})=y_{0},$ $(y_{0}, t_{0})\in\Gamma_{T}$, continuous,

(19) surjective and non-increasing in $\gamma$ for each $t\fbox{Error::0x0000},$

non-decreasing in $t$for each

$\gamma$

},

where, we define atotal order $\succeq$ on the initial/boundaryset $\Gamma_{T}$ by

Let $\theta\in\Theta_{T}$

.

For each $w\in W$ and _$z\in[0$, 1) define

$\omega=\omega_{\theta,w,z}$,

a

non-negative valued

continuous function of $(s, t)$ satisfying$0\leqq s\leqq t\leqq T$, by

(21) $\omega_{\theta,w,z}(s, t)=\{\begin{array}{l}w(\theta((z, 0), t), t) , if s=0,w(\theta((O, s), t), t) , if s>0.\end{array}$

Let $\{N_{\theta,w,z}|z\in[0, 1), w\in W\}$ be a set ofprocesses, with each$N_{\theta,w,z}$ being apoint process $N$

introduced in

\S

1.2 with the intensity function in (16) determined by$\omega=\omega_{\theta,w,z}.$

The quantity in (17) for the choice (21) is

(22) $\Omega_{\theta,w,z}(0, t)=\int_{0}^{t}w(\theta((z, 0), u), u)du, and\Omega_{\theta,w}(s, t)=\int_{S}^{t}w(\theta((0, s), u), u)du.$

Let $\mu 0$ be as in Theorem 1, and define a function $\varphi_{\theta}(dw, \gamma, t)$ on $(\gamma, t)\in\Delta_{T}$ taking values

in the

measures

on $W$, by

(23) $\varphi_{\theta}(dw, \gamma, t)=\int_{z\in[y_{0},1)}P[N_{\theta,w,z}(t)=N_{\theta,w,z}(t_{0})]\mu_{0}(dw\cross dz) , \gamma=(y_{0}, t_{0})\in\Gamma,$ and define

a

map $G:\Theta_{T}arrow\Theta_{T}$ by

(24) $G(\theta)(\gamma, t)=1-\varphi_{\theta}(W, \gamma, t) , \gamma=(y_{0}, t_{0})\in\Gamma, (\gamma, t)\in\Delta_{T}.$

Theorem 2 ([2]) The

_function

$yc$ in Theorem 1 is the unique

fixed

point

of

$G$ in (24), and

$\mu_{t}$ in Theorem 1 is a measure valued junction

of

$(\gamma, t)\in\Delta_{T}$ uniquely determined by $\mu_{t}(dw\cross$

$[yc(\gamma, t), 1))=\varphi_{y_{C}}(dw, \gamma, t)$, where $\varphi_{y_{C}}$ is obtained by the substitution $\theta=yc$ in (23). ◇

See [2] for proofsofTheorem 1 and Theorem 2.

Explicit formula of$G$in (24) isfound using theproperties of the process introduced in

_\S

1.2;

(25) $G( \theta)((y_{0},0), t)=1-\int_{W\cross[y0,1)}e^{-\Omega_{\theta,w,z}(0,t)}\sigma(w, z) \lambda(dw)dz,$

for $\gamma=(y_{0},0)\in\Gamma_{i}$, and for $\gamma=(0, t_{0})\in\Gamma_{b}\cap\Gamma_{t}$

$G( \theta)((0, t_{0}), t)=1-\int_{W\cross[0,1)}e^{-\Omega_{\theta,w,z}(0,t)}\sigma(w, z)$$\lambda(dw)dz$

(26) $- \int_{W\cross[0,1)}\sum_{k\geqq 1}\int_{0\leqq u_{1}\leqq\cdots\leqq u_{k}\leqq t_{0}}w(\theta((z, 0), u_{1}), u_{1})e^{-\Omega_{\theta,w,z}(0,u_{1})}$

$\cross\prod_{i=2}^{k}(w(\theta((0, u_{i-1}), u_{i}), u_{i})e^{-\Omega_{\theta,w}(u_{t-1_{\rangle}}u_{i}))}\cross e^{-\Omega_{\theta,w}(u_{k_{\rangle}}t)}\prod_{i=1}^{k}du_{i}\sigma(w, z)\lambda(dw)dz.$

2

Open problems.

I came across several questions, while trying to generalize [3] to [2]. I welcome answers!

2.1

Lipschitz

continuity condition

and

uniqueness

of solution.

In the beginning we assumed bounded Lipschitz constant condition (3) for the intensity

func-tions, in addition to (2). The latter is a natural condition for (14) to make sense, while the

Theorem 3 ([2]) Under the condition (2) and

(27) $C_{W}’:= \sup \sup |w(y, t)-w(y’, t <\infty,$

$w\in W(y,t) , (y’,t’)\in[O, 1]\cross[0,T]$

replacing (3), the map $G:\ominus\tauarrow\ominus\tau$

defined

by (25) and (26) has a

fixed

point. ◇

The notations in the definition of$G$ areintroduced in (13), (12), (4), (5), (6), (19), and (22). Note that $G(\theta)\in\Theta_{T}$ holds with theassumption (27) which isweaker than (3). It is easy to see

that $\ominus\tau\subset C^{0}(\triangle\tau;[0,1])$ isa bounded, closed, and convexset. According to the Schauder fixed

point theorem, compactness of$G$ : $\Theta_{T}arrow\Theta_{T}$ thereforeimplies Theorem 3. Since$C^{0}(\triangle\tau;[0,1])$

is a bounded set with respect to the supremum norm, the Arzela-Ascoli theorem implies that

the following implies compactness of$G$, under the conditions as in Theorem 3. Lemma 4 (i) The map $G:\Theta_{T}arrow\Theta_{T}$ is continuous, and

(ii) the

_functions

in the image set$G(\ominus\tau)$ are equicontinuous. ◇

A proof of Lemma4 is in [2, Appendix].

In view ofTheorem 3, the assumption (3) would be more crucial for the uniqueness of the solution than for the existence. Considering the fluid picture, breakdown of uniqueness, if itever

does,wouldlikelytooccurfor

an

$(unstable’$ _{initialcondition, that}$is, the$evaporationrate $w(y, t)$

is increasing in $y$, so that once a small portion of the fluid components begin to move in the

downstream direction (larger $y$), the amount of evaporation increases. (Turning the argument

in the other direction, I conjecture that if $W$ contains only those functions _{$w(y, t)$} which are

non-increasing in $y$, then we have aproof of uniqueness without the assumption (3).) We give

a candidateinitial condition. We will focus on uniqueness of thefixed point for (25). I have no

idea on how to deal with (26).

Fig. 1

Fix $\delta\in(0, \frac{1}{2})$, and let $w_{a}^{*}$ and $A_{a}$ be positive valued measurable functions of$a\in[\delta$,1$]$. For

the probability space $(W, \mathcal{B}(W), \lambda)$ we choose $W=\{w_{a}|a\in\{0\}\cup[\delta$, 1 with

(28) $w_{0}(y, t)=0$_{, and} $w_{a}(y, t)=w_{a}^{*}f( \frac{y-a}{A_{a}})$, $\delta\leqq a\leqq 1,$

and

(29) $\lambda(dw_{a})=1_{[\delta,1]}da+\delta\delta_{0}(da) , a\in\{0\}\cup[\delta, 1$

where $\delta_{0}$ is the unitmeasure concentratedon $0$, and $f$ : $\mathbb{R}arrow[0$, 1$]$ isa $C^{1}$ functionsatisfying

and weperformed achange in the integrationvariable$w\mapsto a$ by_{$w=w_{a}$}

.

We choose the density function$\sigma_{a}(y)=\sigma(w_{a}, y)$ of the initial data definedby $\sigma_{a}(y)dy\lambda(dw_{a})=\mu_{0}(da\cross dy)$ , as

(31) $\sigma_{a}(y)=\{\begin{array}{ll}\frac{1}{\delta_{1}}\phi(\frac{1}{\delta}(y-a+\delta \delta\leqq a\leqq 1,\overline{\delta^{2}}(\delta-y)_{+}+\frac{1}{\delta^{2}}(y-(1-\delta))_{+}, a=0,\end{array}$

where$\phi(z)=\{$ 1,

$0\leqq z<1$,

and we used anotation$x+:= \frac{1}{2}(x+|x|)$. The assumptions

$0,$ $z<0$

or

$z\geqq 1,$

(13), (12), (2), and $\lambda(W)=1$ hold, if

(32) $M_{w}:= \int_{\delta}^{1}w_{a}^{*}da<\infty.$

Substituting (29) and (31) in (25), and using (22), we have for $0\leqq y_{0}<1$ and $0\leqq t\leqq T,$

$y_{C}((y0,0), t)=G(yc)((y_{0},0), t):=I_{1}(y_{0}, t)-I_{2}(y_{C})(y_{0}, t)$, where

(33)

$I_{1}(y_{0}, t)=\{\begin{array}{ll}1-\delta+y_{0}-\frac{1}{2\delta}y_{0}^{2}, 0\leqq y_{0}<\delta,\frac{1}{2}, \delta\leqq y_{0}\leqq 1-\delta,y_{0}+\frac{1}{2\delta}(1-y_{0})^{2}, 1-\delta<y_{0}<1,\end{array}$

$I_{2}(y_{C})(y_{0}, t$$)= \int_{y0}^{1}dz\int_{z\vee\delta}^{1\wedge(z+\delta)}dae$

$-w_{a}^{*} \int_{0}^{t}f(\frac{1}{A_{a}}(yc((z, 0), u)-a)du_{dz}.$

It follows from$w_{a}(z)\sigma_{a}(z)=0$for all $z$ and$a$ that there isa constant flow$y_{C}((y_{0},0), t)=y_{0}$

for all $y0$ and $t$. The problem is to decide if there are $\delta\in(0, \frac{1}{2})$, $A_{a},$ $w_{a}^{*}$, and $f$, satisfying (32) and (30) such that (33) holds for anon-constant flow, continuously increasing in $t$ and

$y_{0}.$

Suppose we

are

on the side that such fixed point $yc$ exists, how could

we

prove it? $A$

standard way is to apply the Schauder’s fixed point theorem as in Theorem 3, with set of functions (domainof $G$) $\Theta_{T}$ replaced by $\tilde{\Theta}_{T}$

which excludes the constant flow. A candidate set might perhaps look like

(34) $\overline{\Theta}_{T}:=\{\theta:\in\Theta_{T}|\theta(y, t)\geqq y+\epsilon t, t\leqq\epsilon(1-y)^{M}, 0\leqq y<1\},$

for some $\epsilon>0$ and $M>0$. The set (34) excludes the constant flow, is a bounded, closed, and

convex

set. Since we

are

restricting the domain of$G$, the compactness ofthe map $G$ stated in Lemma 4 is preserved, if (27) holds. This requires boundedness of $w_{a}^{*}$, so we may

as

well put

$w_{a}^{*}=1,$ $\delta\leqq a\leqq 1$. If this works, it only remains to prove, for

some

set of$\delta,$ $A_{a},$ $f,$ $\epsilon,$ $M,$

(35) $G(\tilde{\Theta}_{T})\subset\overline{\Theta}_{T}$ $($?$)$.

2.2

Stationary

solution.

The boundary condition (8) is designed as a necessary condition for (10) to have a stationary

solution for $\mu_{t}(dw,$$[y,$$1$ in the

case

when $W$ is supported on intensity functions $w$ which are

$\partial\mu_{t}$

constant in $t$. Here, a stationary solution means a solution satisfying

$\overline{\partial t}=0,$ $t\in[0, T]$, in

(10). Dropping $t$ from the notations $w(y, t)$ and replacing $\mu_{t}$ by $\mu_{0}$ and using (11), we have

Differentiating by $t$, and using (14) with (11), and finally replacing

$y_{C}((y_{0}, t_{0}), t)$ by $y$, we have

(36) $( \int_{W\cross[y,1)}w(z)\sigma(w, z)\lambda(dw)dz)\sigma(w, y)$ $= \int_{y}^{1}w(z)\sigma(w, z)$$dz,$ $(w, y)$ $\in W\cross[0$,1$)$

.

Thequestionisto decide whether (36) has

a

solution$\sigma$ : $W\cross[O, 1$) $arrow[0, \infty$)which ismeasurable

and satisfies (13) and (12). For simplicity, we

assume

in the following

(37) $\inf w(y)>0, w\in W.$

$y\in[0,1]$

We can reduce the unknown from a function $\sigma$ on _$W\cross[O$,1$]$ to that on $[0$,1$]$, as follows. Put

(38) $\tilde{u}(w, y)=\int_{y}^{1}w(z)\sigma(w, z)dz$ and $v(y)= \int_{W}\tilde{u}(w, y)\lambda(dw)$.

Then (36) implies $\frac{\tilde{u}’(w,y)}{\tilde{u}(w,y)}=-\frac{w(y)}{v(y)}$ which further implies

(39)

$\tilde{u}(w, y)=C(w)e^{-}\int_{0}^{y}\frac{w(z)}{v(z)}dz$

for

some

function $C:Warrow[O, \infty$_{). With (38)} we _{further have}

(40) $v(y)= \int_{W}C(w)e^{-\int_{0}^{y}\frac{w(z)}{v(z)}dz}\lambda(dw)$_.

A boundary condition $\tilde{u}(w, 1)=0$ is implied from _{(38), which, with (37), further implies}

(41) $v(1-)=0.$

Note that (38), (39), and (40) imply$\int_{W}\sigma(w, y)d\lambda(dw)=1$, sothat (13) holds. On the other

hand, (12) implies with a similar argument,

(42) $C(w)=( \int_{0}^{1}\frac{1}{v(z)}e^{-\int_{0}^{z}\frac{w(x)}{v(x)}dx}dz)^{-1}$ This and (40) imply

(43)

$v(y)=H(v)(y):= \int_{W}\frac{e^{-\int}\lambda(dw)}{\int_{0}^{1_{\frac{1}{v(z}}\frac{(x)}{(x)}dx_{dz}}}, y\in[0,1)$.

We will define $H(v)(1)$ $:=0$ _to

save

_{any further remarks on compactness issues at $y=1$. If}

we have a non-increasing, continuous, non-negative function $v$ : $[0, 1$) $arrow[0, \infty$) which satisfies

(43) and (41), then (43) and (42) determine (39) and $\sigma$ is given by differentiating (38), which

satisfies (36), (13), and (12).

Note that (42), (1), and (37) imply $\frac{1}{C(w)}=\int_{0}^{1}\frac{1}{v(z)}e^{-\int_{0}^{z}\frac{w(x)}{v(x)}dx}dz\geqq\frac{1}{\Vert w\Vert_{T}}$

. This with

(43) and (2) further imply

Let $D$ be the set of non-increasing, continuous functions $f$ : $[0, 1]arrow[0, M_{W}]$, satisfying

$f(1)=0$. Then $D$ is bounded, closed, and convex. Moreover, (43) and (44) imply $H(D)\subset D.$

Therefore, if we can prove the compactness of the map $H$, namely, the continuity ofthe map $H$ and the equicontinuity of the functions in the image set $H(D)$_{, the Schauder’s fixed} point theorem implies existence of$v$, and consequently, of a stationary solutionto the original PDE.

If$W$issupportedonconstant $w’ s$_, then there is astationarysolution to theoriginal problem.

For $w\in W$ let $w^{*}\in[0, \infty$) $\backslash \{O\}$ be its (constant) value. Then (42) and (41) imply

(45) $\frac{1}{C^{*}(w^{*})}:=\frac{1}{C(w)}=-\frac{1}{w^{*}}\int_{0}^{1}\frac{d}{dz}(e^{-w^{*}\int_{0}^{z}\frac{1}{v(x)}dx})dz=\frac{1}{w}*,$ and (40) further implies, with$\lambda^{*}(dw^{*})=\lambda(dw)$ the image

measure

of $\lambda$

, supported on $[0, \infty$),

$v(y)=-v(y) \frac{d}{dy}(\int_{0}^{\infty}e^{-w^{*}\int_{0}^{y}\frac{dz}{v(z)}}\lambda^{*}(dw^{*}))$ . Dividing by $v(y)$ and integrating, we have a formula which is expressed as

(46) $\int_{0}^{y}\frac{dz}{v(z)}=\varphi^{-1}(1-y)$,

where $\xi=\varphi^{-1}(x)$ is the inverse function of$\varphi(\xi)=\int_{0}^{\infty}e^{-w^{*}\xi}d\lambda^{*}(dw^{*})$, $\xi\geqq 0.$

Since we assume (37) in this subsection, we have $\varphi(0)=1$ and $\varphi(\infty)=0$,

so

that

(47) $\varphi^{-1}(0)=\infty$ and $\varphi^{-1}(1)=0.$

(41) and (37) imply that (47) is consistent with (46). Substituting (46) and (45) in (39), we

have $\tilde{u}^{*}(w^{*}, y)$ $:=\overline{u}(w, y)=w^{*}e^{-w^{*}\varphi^{-1}(1-y)}$

.

Substituting this in (38), we arrive at $\sigma^{*}(w^{*}, y):=$

$\sigma(w, y)=-\frac{d}{dy}(e^{-w\varphi^{-1}(1-y)})$ and$\mu(dw\cross[y, 1))=\lambda(dw)\int_{y}^{1}\sigma(w, z)$$dz=e^{-w\varphi^{-1}(1-y)}\lambda(dw)$

.

Thus for the constant $w$ case, we have an explicit formula for the solution $\mu$ in terms of the

generating functionof$\lambda.$

References

[1] A. Bressan, Hyperbolic systems

_of

conservation laws, OxfordUniv. Press, Oxford, 2005.

[2] T. Hattori, Point process with last-arrival-time dependent intensity and 1-dimensional

incompressible

_fluid

system with evaporation, preprint (2014) http:$//$_{arxiv.$org/abs/$}

1409.5117.

[3] T. Hattori, S. Kusuoka, Stochastic rankingprocess with space-time dependent

iniensities,

ALEA, Lat. Am. J. Probab. Math. Stat. 9 (2) (2012) 571-607. Laboratory ofMathematics, Faculty of Economics, Keio University,

Hiyoshi Campus, 4-1-1 Hiyoshi, Yokohama 223-8521, Japan.

URL: http:$//web$

.

_econ.keio.ac.jp/staff/hattori/research.htm

email: [email protected]